Boil potatoes in salted water to cover until tender, about 10 minutes. Drain the potatoes, reserving 1 ½ cups of the cooking liquid, and mash. Measure 1 Cup of the mashed potatoes and reserve the rest for another use.
Stir together the mashed potatoes, margarine, ½ cup of the sugar and salt. Cool to lukewarm and add yeast, eggs and the reserved 1 ½ cups cooking liquid. Mix well. Add flour and stir. Knead on a lightly floured surface until the dough is smooth and elastic. Set in a lightly oiled bowl, cover with a damp cloth and let rise in a warm place until doubled in bulk. Punch down and refrigerate until chilled, about 1 hour.
Butter three 9-inch round or square baking dishes. Distribute brown sugar evenly among the pans. Sprinkle with enough water to make the sugar very wet. Distribute walnuts evenly over the brown sugar
Roll out the dough into a ½-inch-thick rectangle on a well-floured surface. Brush with melted butter. Stir together the remaining 2 ¼ cups sugar and cinnamon. Sprinkle over the buttered dough. Roll up the dough as you would a jelly roll. Cut into ½-inch-thick slices. Arrange the slices cut-side up in the prepared pans so that they are almost touching. Cover and let rise in a warm place until doubled in bulk, about 1 hour.
Preheat oven to 350 degrees F.
Bake the sticky buns until golden brown, 20 to 30 minutes.
Immediately invert the buns onto a plate.
MAKES 6 TO 8 STICKY BUNS PER PAN